USING CUBIC SPLINES TO SOLVE SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS: A TEACHING APPLICATION

F. Giménez Palomares, J.F. Giménez Luján, J.A. Monsoriu Serrá, P. Fernández de Córdoba Castellá

Universitat Politècnica de València (SPAIN)
A lot of complex real-life problems are modeled from ordinary differential equations and partial derivatives. When the initial conditions of the problem refer to the border of a domain in which it is desired to calculate the solution, we speak of a boundary problem. Particularly interesting, due to their multiple applications, are ordinary linear differential equations of second order with boundary conditions where the values of the solution sought at the extremes of an interval (Dirichlet conditions) or those of their derivatives are assumed (Neuman conditions).

Cubic splines, together with the technique of least squares, allow to efficiently solve this type of problems. Other widely used methods are the so-called shooting method (the approximate solution is obtained from the first-order system associated with the second-order equation, the bisection method for solving non-linear equations, and the use of classical methods) and finite differences methods.

We present here a teaching tool designed as a graphical user interface of Matlab, that allows to interactively visualize the procedure of the cubic splines and to compare them with the other methods or with the exact solution if it is known. With this tool the students can more easily assimilate the concepts related to the use of cubic splines to solve ordinary linear differential equations of second order with boundary conditions and his properties. We propose the following study methodology: 1) introduction of the necessary theory on numerical methods for solving differential equations in the usual classes, 2) realization of a practice in computer rooms where students learn to use the virtual laboratories related.

This teaching experience has been carried out in the subject Mathematical Methods of the Bachelor's Degree in Industrial Engineering of the Polytechnic University of Valencia during the last two courses, having found a very good reception by our students and an optimal use as shown by the academic results obtained.
keywords: differential equation, cubic spline, shoting method, finite differences method, virtual laboratory, matlab, graphical interface, active methodologies.